In
image processing, a Gabor filter, named after
Dennis Gabor, who first proposed it as a 1D filter.[1]
The Gabor filter was first generalized to 2D by Gösta Granlund,[2] by adding a reference direction.
The Gabor filter is a
linear filter used for
texture analysis, which essentially means that it analyzes whether there is any specific frequency content in the image in specific directions in a localized region around the point or region of analysis. Frequency and orientation representations of Gabor filters are claimed by many contemporary vision scientists to be similar to those of the
human visual system.[3] They have been found to be particularly appropriate for texture representation and discrimination. In the spatial domain, a 2D Gabor filter is a
Gaussiankernel function modulated by a
sinusoidalplane wave (see
Gabor transform).
In this equation, represents the wavelength of the sinusoidal factor, represents the orientation of the normal to the parallel stripes of a
Gabor function, is the phase offset, is the sigma/standard deviation of the Gaussian envelope and is the spatial aspect ratio, and specifies the ellipticity of the support of the Gabor function.
Wavelet space
Gabor filters are directly related to
Gabor wavelets, since they can be designed for a number of dilations and rotations. However, in general, expansion is not applied for Gabor wavelets, since this requires computation of bi-orthogonal wavelets, which may be very time-consuming. Therefore, usually, a filter bank consisting of Gabor filters with various scales and rotations is created. The filters are convolved with the signal, resulting in a so-called Gabor space. This process is closely related to processes in the primary
visual cortex.[8]
Jones and Palmer showed that the real part of the complex Gabor function is a good fit to the receptive field weight functions found in simple cells in a cat's striate cortex.[9]
Time-causal analogue of the Gabor filter
When processing temporal signals, data from the future cannot be accessed, which leads to problems if attempting to use Gabor functions for processing real-time signals that depend on the temporal dimension. A time-causal analogue of the Gabor filter has been developed in [10] based on replacing the Gaussian kernel in the Gabor function with a time-causal and time-recursive kernel referred to as the time-causal limit kernel. In this way, time-frequency analysis based on the resulting complex-valued extension of the time-causal limit kernel makes it possible to capture essentially similar transformations of a temporal signal as the Gabor filter can, and as can be described by the Heisenberg group, see [10] for further details.
Extraction of features from images
A set of Gabor filters with different frequencies and orientations may be helpful for extracting useful features from an image.[11] In the discrete domain, two-dimensional Gabor filters are given by,
where B and C are normalizing factors to be determined.
2D Gabor filters have rich applications in image processing, especially in
feature extraction for texture analysis and segmentation.[12] defines the frequency being looked for in the texture. By varying , we can look for texture oriented in a particular direction. By varying , we change the support of the basis or the size of the image region being analyzed.
Applications of 2D Gabor filters in image processing
In document image processing, Gabor features are ideal for identifying the script of a word in a multilingual document.[13] Gabor filters with different frequencies and with orientations in different directions have been used to localize and extract text-only regions from complex document images (both gray and colour), since text is rich in high frequency components, whereas pictures are relatively smooth in nature.[14][15][16] It has also been applied for facial expression recognition [17]
Gabor filters have also been widely used in pattern analysis applications. For example, it has been used to study the directionality distribution inside the porous spongy
trabecularbone in the
spine.[18] The Gabor space is very useful in
image processing applications such as
optical character recognition,
iris recognition and
fingerprint recognition. Relations between activations for a specific spatial location are very distinctive between objects in an image. Furthermore, important activations can be extracted from the Gabor space in order to create a sparse object representation.
importnumpyasnpdefgabor(sigma,theta,Lambda,psi,gamma):"""Gabor feature extraction."""sigma_x=sigmasigma_y=float(sigma)/gamma# Bounding boxnstds=3# Number of standard deviation sigmaxmax=max(abs(nstds*sigma_x*np.cos(theta)),abs(nstds*sigma_y*np.sin(theta)))xmax=np.ceil(max(1,xmax))ymax=max(abs(nstds*sigma_x*np.sin(theta)),abs(nstds*sigma_y*np.cos(theta)))ymax=np.ceil(max(1,ymax))xmin=-xmaxymin=-ymax(y,x)=np.meshgrid(np.arange(ymin,ymax+1),np.arange(xmin,xmax+1))# Rotationx_theta=x*np.cos(theta)+y*np.sin(theta)y_theta=-x*np.sin(theta)+y*np.cos(theta)gb=np.exp(-0.5*(x_theta**2/sigma_x**2+y_theta**2/sigma_y**2))*np.cos(2*np.pi/Lambda*x_theta+psi)returngb
^
Gabor, D. (1946). "Theory of communication". J. Inst. Electr. Eng. 93.
^
Granlund G. H. (1978). "In Search of a General Picture Processing Operator". Computer Graphics and Image Processing. 8 (2): 155–173.
doi:
10.1016/0146-664X(78)90047-3.
ISSN0146-664X.
^Haghighat, M.; Zonouz, S.; Abdel-Mottaleb, M. (2013). "Identification Using Encrypted Biometrics". Computer Analysis of Images and Patterns. Lecture Notes in Computer Science. Vol. 8048. pp. 440–448.
doi:
10.1007/978-3-642-40246-3_55.
ISBN978-3-642-40245-6.
^S Sabari Raju, P B Pati and A G Ramakrishnan, “Text Localization and Extraction from Complex Color Images,” Proc. First International Conference on Advances in Visual Computing (ISVC05), Nevada, USA, LNCS 3804, Springer Verlag, Dec. 5-7, 2005, pp. 486-493.